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Vectors
VECTOR BASICS – What is a vector? Vector Components Graphically, a vector is represented by an arrow, defining the direction, and the length of the arrow defines the vector's magnitude. Point T is the origin and it is drawn to point D. This vector can be represented as TD. The arrow above the letters acknowledges that this quantity is a vector. A vector is a quantity which has magnitude and direction. Displacement, velocity, acceleration, and force are examples of the vector quantities. Vectors can be of any dimension, but are most commonly seen in 2 or 3 dimensions. Vectors are directed in two dimensions - upward and rightward, northward and westward, eastward and southward, etc. A vector of dimension 3 can represent a physical quantity which is directional such as positionhttp://www.euclideanspace.com/physics/kinematics/position/index.htm, velocityhttp://www.euclideanspace.com/physics/kinematics/velocity/index.htm , accelerationhttp://www.euclideanspace.com/physics/kinematics/acceleration/index.htm, forcehttp://www.euclideanspace.com/physics/statics/forces/index.htm, momentumhttp://www.euclideanspace.com/physics/dynamics/linear/index.htm, etc. For example if the vector represents a point in space, these 3 numbers represent the position in the x, y and z coordinates (See coordinate systems). Where x, y and z are mutually perpendicular axis in some agreed direction and units. A 3 dimensional vector may also represent a displacement in space, such as a translation in some direction. Vectors directed at angles In situations in which vectors are directed at angles to the coordinate axes, a useful mathematical trick will be employed to transform the vector into two parts, with each part being directed along the coordinate axes. For example, a vector which is directed northwest can be thought of as having two parts - a northward and a westward part. A vector which is directed upward and rightward can be thought of as having two parts - an upward and a rightward part. Any vector directed in two dimensions can be thought of as having an influence in two different directions. That is, it can be thought of as having two parts. Each part of a two-dimensional vector is known as a component. The components of a vector depict the influence of that vector in a given direction. The combined influence of the two components is equivalent to the influence of the single two-dimensional vector. The single two-dimensional vector could be replaced by the two components. If Fido's dog chain is stretched upward and rightward and pulled tight by his master, then the tensional force in the chain has two components - an upward and a rightward component. To Fido, the influence of the chain on his body is equivalent to the influence of two chains on his body - one pulling upward and the other pulling rightward. If the single chain were replaced by two chains (each one having the magnitude and direction of the components), the Fido would not know the difference. This is not because Fido is dumb (a quick glance at his picture reveals that he is certainly not that), but rather because the combined influence of the two components is equivalent to the influence of the single two-dimensional vector. Combining Vectors Vector Addition Adding two vectors is equivalent to putting the tail of one vector against the head of the other. This is called head-to-tail addition. To do this place the start of the arrow b at the tip of the arrow a, and then drawing an arrow from the start of a to the tip of b. The new arrow drawn represents the vector a + b, as illustrated below: Vector Subtraction Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the ends of a and b at the same point, and then draw an arrow from the tip of b to the tip of a. That arrow represents the vector a − b, as illustrated below: Vectors created from different forces Head-to-tail Method Now we will see how the head to tail method of vector additionhttp://www.physicsclassroom.com/Class/vectors/u3l1b.html#head applies to situations involving the addition of force vectors. A force board (or force table) is a common physics lab apparatus that has three (or more) strings or cables attached to a center ring. The strings or cables exert forces upon the center ring in three different directions. Typically the experimenter adjusts the direction of the three forces, makes measurements of the amount of force in each direction, and determines the vector sum of three forces. Suppose that a force board or a force table is used such that there are three forces acting upon an object (the object is the ring in the center of the force board or force table). In this situation, each of the three forces are acting in two-dimensions. A top view of these three forces could be represented by the diagram above. The goal of a force analysis is to determine the net forcehttp://www.physicsclassroom.com/Class/newtlaws/u2l2d.html (and the corresponding acceleration). The net force is the vector sum of all the forces. That is, the net force is the resultant of all the forces; it is the result of adding all the forces as vectors. For the situation of the three forces on the force board, the net force is the sum of force vectors A + B + C. One method of determining the vector sum of these three forces (i.e., the net force") is to employ the method of head-to-tail addition. In this method, an accurately drawn scaled diagram is used and each individual vector is drawn to scale. Where the head of one vector ends, the tail of the next vector begins. Once all vectors are added, the resultant (i.e., the vector sum) can be determined by drawing a vector from the tail of the first vector to the head of the last vector. This procedure is shown below. The three vectors are added using the head-to-tail method. Incidentally, the vector sum of the three vectors is 0 Newtons - the three vectors add up to 0 Newtons. The purpose of adding force vectors is to determine the net force acting upon an object. In the above case, the net force (vector sum of all the forces) is 0 Newtons. This would be expected for the situation since the object (the ring in the center of the force table) was at rest and staying at rest. We would say that the object was at equilibrium. Any object upon which all the forces are balanced (Fnet = 0 N) is said to be at equilibrium. Quite obviously, the net force is not always 0 Newtons. In fact, whenever objects are accelerating, the forces will not balance and the net force will be nonzero. This is consistent with Newton's first law of motion. PRACTICE PROBLEMS ---- 1. A pack of five Artic wolves are exerting five different forces upon the carcass of a 500-kg dead polar bear. A top view showing the magnitude and direction of each of the five individual forces is shown in the diagram at the right. The counterclockwise convention is used to indicate the direction of each force vector. Remember that this is a top view of the situation and as such does not depict the gravitational and normal forces (since they would be perpendicular to the plane of your computer monitor); it can be assumed that the gravitational and normal forces balance each other. Use a scaled vector diagram to determine the net force acting upon the polar bear. Then compute the acceleration of the polar bear (both magnitude and direction). 2. If John is pulling Vida with 8 N of force North, and Tom is pulling with a force of 5 N south, in which direction will Vida move? 3. A new TV is inside a box in the hallway. Antoine, Sam, Jeff and Shawn see the TV and try to push it towards the elevator to take it into Shawn’s car. Antoine pushes with 5 N to wards the East. Jeff pushes with 4 Newtons to the west, Shawn pushes with 1 Newton to the South, and Sam pushes with 3 Newtons to the North. In which directions will the box move? ANSWERS 1. The Net Force is 39.4N at 324 degrees The acceleration of the polar bear is 0.0788 m/s/s. This value is found by dividing Fnet by m (39.4N/500kg). 2. Since John is applying the greater force, 8N north – 5N south, = 3N to the North. Vida will move towards the north. 3. Do this step by step: 5N(East)-4N(West)=1N(East) 3N(North)-1N(South)=2N(North) You are left with 1N going East and 2N going North. The box will travel North East with a force equal to the square root of 5. REFERENCES ---- 1. http://www.physicsclassroom.com/Class/vectors/U3L3a.html 2. http://www.glenbrook.k12.il.us/gbssci/Phys/Class/vectors/u3l1a.html 3. http://id.mind.net/~zona/mstm/physics/mechanics/vectors/vectors.html 4. http://www.gamedev.net/reference/articles/article1832.asp 5. http://www.euclideanspace.com/maths/algebra/vectors/ 6. "Vector (spatial)" Wikipedia, The Free Encyclopedia. Wikimedia Foundation, Inc. June 9, 2006. . Resources ---- 1. http://www.glenbrook.k12.il.us/gbssci/Phys/Class/vectors/u3l1a.html © Tom Henderson, 1996-2004 2. http://en.wikipedia.org/wiki/Vector_(spatial) 3. http://www.physicsclassroom.com/about.html#credits © 2004 The Physics Classroom and Mathsoft Engineering & Education, Inc. 4. Gewirtz, Herman. Barron’s How to Prepare for the Sat II Physics. 8th ed. Barron's Educational Series, Inc., 2004. 5. Zitzewitz, Paul W. "Glencoe Physics: Principles and Problems". Copyright © 1999 by The McGraw-Hill Companies, Inc. June 5, 2006. IMAGES 6. http://en.wikipedia.org/wiki/Vector_(spatial) Illustration of two very slightly different methods of adding vectors a and b. 7. http://en.wikipedia.org/wiki/Vector_(spatial) Diagram illustrating the subtraction a−b of vectors a and b. 8. http://www.physicsclassroom.com/Class/vectors/U3L3a.html Illustration of a force board. resume writing services